Inventiones mathematicae

, Volume 97, Issue 1, pp 195–207 | Cite as

The algebraic characterization of the exteriors of certain 2-knots

  • Jonathan A. Hillman


We study 2-knots with virtually solvable group by applying recent work of Freedman to the 4-manifolds obtained by surgery on such knots. In particular we show that “Gluck reconstruction” is the only ambiguity in recovering the Cappell-Shaneson knots (as TOP locally flat knots) from their groups alone.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bak, A.: The computation of even dimensional surgery groups of odd torsion groups. Commun. Algebra6, 1393–1458 (1978)Google Scholar
  2. 2.
    Bieri, R.: Homological dimensions of discrete groups. Queen Mary College Lecture Notes, London 1976Google Scholar
  3. 3.
    Browder, W.: Diffeomorphisms of 1-connected manifolds. Trans. Am. Math. Soc.128, 155–163 (1967)Google Scholar
  4. 4.
    Cappell, S.E.: Mayer-Vietoris sequences in HermiteanK-theory. In: Bass, H. (ed.) HermiteanK-theory and geometric applications (Lect. Notes Math. vol. 343, pp. 478–512) Berlin Heidelberg New York: Springer 1973Google Scholar
  5. 5.
    Cappell, S.E.: A splitting theorem for manifolds. Invent. Math.33, 69–170 (1976)Google Scholar
  6. 6.
    Cappell, S.E., Shaneson, J.L.: There exist inequivalent knots with the same complement. Ann. Math.103, 349–353 (1976)Google Scholar
  7. 7.
    Cappell, S.E., Shaneson, J.L.: On 4-dimensionals-cobordisms. J. Differ. Geom.22, 97–115 (1985)Google Scholar
  8. 8.
    Cochran, T.: Ribbon knots inS 4. J. Lond. Math. Soc.28, 563–576 (1983)Google Scholar
  9. 9.
    Culler, M.C., Gordon, C.McA., Luecke, J., Shalen, P.B.: Dehn surgery on knots. Ann. Math.125, 237–300 (1987)Google Scholar
  10. 10.
    Davis, M.W.: Groups generated by reflections and aspherical manifolds not covered by Euclidean space. Ann. Math.117, 293–324 (1982)Google Scholar
  11. 11.
    Dyer, E., Vasquez, A.T.: The sphericity of higher dimensional knots. Can. J. Math.25, 1132–1136 (1972)Google Scholar
  12. 12.
    Farrell, F.T., Hsiang, W.C.: The Whitehead group of poly-(finite or cyclic) groups. J. Lond. Math. Soc.24, 308–324 (1981)Google Scholar
  13. 13.
    Farrell, F.T., Hsiang, W.C.: Topological characterization of flat and almost flat riemannian manifoldsM n(n≠3, 4). Am. J. Math.105, 641–672 (1983)Google Scholar
  14. 14.
    Fox, R.H.: A quick trip through knot theory. In: M.K. Fort, Jr. (ed.) Topology of 3-manifolds and related topics. Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 120–167Google Scholar
  15. 15.
    Freedman, M.H.: The topology of four-dimensional manifolds. J. Differ. Geom.17, 357–453 (1982)Google Scholar
  16. 16.
    Freedman, M.H.: The disc theorem for four-dimensional manifolds. Proc. Int. Congr. Math., Warsaw 1983, pp. 647–663Google Scholar
  17. 17.
    Gildenhuys, D.: Classification of soluble groups of cohomological dimension two. Math. Z.166, 21–25 (1979)Google Scholar
  18. 18.
    Gluck, H.: The embedding of two-spheres in the four-sphere. Trans. Am. Math. Soc.104, 308–333 (1962)Google Scholar
  19. 19.
    Gordon, C.McA.: Knots in the 4-sphere. Comment. Math. Helv.51, 585–596 (1976)Google Scholar
  20. 20.
    Hillman, J.A.: High dimensional knot groups which are not two-knot groups. Bull. Aust. Math. Soc.16, 449–462 (1977)Google Scholar
  21. 21.
    Hillman, J.A.: Orientability, asphericity and two-knot groups. Houston J. Math.6, 67–76 (1980)Google Scholar
  22. 22.
    Hillman, J.A.: Abelian normal subgroups of two-knot groups. Comment. Math. Helv.61, 122–148 (1986)Google Scholar
  23. 23.
    Kato, M.: A concordance classification of PL homeomorphisms ofS p ×S q. Topology8, 271–383 (1969)Google Scholar
  24. 24.
    Kervaire, M.A.: Les noeuds de dimensions supérieures. Bull. Soc. Math. Fr.93, 225–271 (1965)Google Scholar
  25. 25.
    Kirby, R.: Problems in low dimensional manifold theory. In: Algebraic and geometric topology. Proc. Symp. Pure Math. 32 vol. 2, Am. Math. Soc. 1978, pp. 273–312Google Scholar
  26. 26.
    Litherland, R.: Topics in knot theory. Thesis, Cambridge University 1978Google Scholar
  27. 27.
    Mihalik, M.: Solvable groups that are simply connected at ∞. Math. Z.195, 79–87 (1987)Google Scholar
  28. 28.
    Papakyriakopolous, C.: On Dehn's lemma and the asphericity of knots. Ann. Math.66, 1–26 (1957)Google Scholar
  29. 29.
    Plotnick, S.P.: The homotopy type of four-dimensional knot complements. Math. Z.183, 447–471 (1983)Google Scholar
  30. 30.
    Plotnick, S.P.: Equivariant intersection froms, knots inS 4, and rotations in 2-spheres. Trans. Am. Math. Soc.296, 543–575 (1986)Google Scholar
  31. 31.
    Plotnick, S.P., Suciu, A.I.: Fibred knots and spherical space forms. J. Lond. Math. Soc.35, 514–526 (1987)Google Scholar
  32. 32.
    Rosset, S.: A vanishing theorem for Euler characteristics. Math. Z.185, 211–215 (1984)Google Scholar
  33. 33.
    Rubinstein, J.H.: Seminar, Canberra, July 1986Google Scholar
  34. 34.
    Stark, C.W.: Structure sets vanish for certain bundles over Seifert manifolds. Trans. Am. Math. Soc.285, 603–615 (1985)Google Scholar
  35. 35.
    Suzuki, S.: Knotting problems of 2-spheres in the 4-sphere. Math. Sem. Notes Kobe Univ.4, 241–371 (1976)Google Scholar
  36. 36.
    Waldhausen, F.: AlgebraicK-theory of generalized free products I, II. Ann. Math.108, 135–256 (1978)Google Scholar
  37. 37.
    Wall, C.T.C.: Poincaré complexes I. Am. Math.86, 213–245 (1967)Google Scholar
  38. 38.
    Wall, C.T.C.: Surgery on compact manifolds. London New York: Academic Press 1970Google Scholar
  39. 39.
    Weinberger, S.: The Novikov conjecture and low-dimensional topology. Comment. Math. Helv.58, 355–364 (1983)Google Scholar
  40. 40.
    Weinberger, S.: On fibering four-and five-manifolds. Israel J. Math.59, 1–7 (1987)Google Scholar
  41. 41.
    Yoshikawa, K.: On two-knot groups with the finite commutator subgroups. Math. Sem. Notes Kobe Univ.8, 321–330 (1980)Google Scholar
  42. 42.
    Hillman, J.A.: 2-knots and their groups. (Austral. Math. Soc. Lecture Series 5) Cambridge New York Melbourne Sydney: Cambridge University Press 1989Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • Jonathan A. Hillman
    • 1
  1. 1.School of Mathematics, Physics, Computing and ElectronicsMacquarie UniversityAustralia

Personalised recommendations